Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #857⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#857⋊C2 |
| unambiguous semidirect product | #857⋊C23 |
| Fitting | C22×#127 ⋊ D3 |
| derived | C42⋊C3 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857⋊C2 |
| commutator, cosocle, hypocenter | C42⋊C3 | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C22×#127 | D3 |
| socle | C24 | C2×S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 | |
2552 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 2552 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219735, #72219748, C2×#11713628, … |
| maximal normal subgroup of | #72219735, #72219748, C2×#11713628, … |
| maximal quotient of | #72214473, C2×#11713628, #72226052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089093);